Re: Review of Mueckenheims book.

In article <1173810471.043981.283420@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
cbrown@xxxxxxxxxxxxxxxxx wrote:

On Mar 13, 9:22 am, Tony Orlow <t...@xxxxxxxxxxxxx> wrote:
cbr...@xxxxxxxxxxxxxxxxx wrote:
On Mar 12, 2:11 pm, Tony Orlow <t...@xxxxxxxxxxxxx> wrote:
mueck...@xxxxxxxxxxxxxxxxx wrote:
On 8 Mrz., 22:46, Tony Orlow <t...@xxxxxxxxxxxxx> wrote:
mueck...@xxxxxxxxxxxxxxxxx wrote:
WM, you don't disagree that there are infinite sets containing just
finite values, such as the reals in [0,1], are you? I certainly agree
that an infinite set of naturals must contain infinite values, but
that's only because they are spaced apart by a unit in value. Isn't tat
your thinking?
If you disregad physical restrictions, then there are infinitely many
real numbers in the interval. Their cardinality, however, is not
larger than "infinite" for any set. Therefore we need no alephs etc.
The binary tree shows that different alephs are self contradictive.
If you take into account the physical restrictions, then there is no
infinite set. And that is the only correct approach.
Regards, WM
Well, since numbers are not physical entities, they don't actually
occupy space on the number line - they are true points. So, between any
two finitely distant points are indeed some infinite number of points.
You say that the only correct approach is to take into account
"physical" restrictions, but where the subject is non-physical, those
restrictions don't exist, though relations do, even if between infinite
nonphysical concepts called numbers.

Wow! That actually made sense for an /entire/ /paragraph/.

:)

DM said, "Wow! That makes even less sense than WM's posts. Although, it
doesn't quite reach the heights of Ross's nonsense."


I think he was referring to the content of the following paragraph.



Where we can count in sequence from one element to any other, that
neighborhood is finite, even if unbounded. Where we can never count
between some pair of objects, such as between, say, ...1111 and
....2222, they are actually infinitely distant elements of a sequence,
since successor() exists.

Ermm... what th'!?!

Cheers - Chas

Was that a question?

One question might be "what is an example of a neighborhood that is
finite yet unbounded?"

In a countable set, there are only a finite number
of elements between any two specific elements.

Counterexample: The set of rationals in [0,1] is a countable set, and
there are an infinite number of elements between any two distinct
elements of that set.

There are an infinite
number of adic numbers between ...111 and ...222, no? And the adics each
have a distinct successor, yes? What was the question?

Another question might be "are you aware of the difference between the
definitions of a list of elements from a set, a total order on the
elements of that set, and a well-ordering of the elements of that
set?"

Cheers - Chas

And it is easy to show that the adics, with their standard ordering, are
not well ordered ( a set being well ordered if and only if every
nonempty subset has a least element.

{...1110, ...1100, ...1000, ... } is an infinite decreasing sequence of
adics under the standard ordering, so the set of all adics is not well
ordered, at least under the standard ordering.
.

Relevant Pages

  • Re: Review of Mueckenheims book.
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  • Re: Review of Mueckenheims book.
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  • Re: Well Ordering the Reals
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  • Re: Well Ordering the Reals
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  • Re: Review of Mueckenheims book.
    ... you don't disagree that there are infinite sets containing just ... finite values, such as the reals in, are you? ... that an infinite set of naturals must contain infinite values, ... If you disregad physical restrictions, ...
    (sci.math)
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